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A current axiomatic definition is that real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field. [d] Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line.
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, as typically construed, states that given two positive numbers and , there is an integer such that .
Least-upper-bound property. Every non-empty subset of the real numbers which is bounded from above has a least upper bound. In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [ 1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the ...
Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number line has a "gap" at each irrational value. In the decimal number system, completeness is equivalent to ...
Assume that a function is defined from a subset of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and −∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities).
Every real number corresponds to a point on the number line. The following paragraph will focus primarily on positive real numbers. The treatment of negative real numbers is according to the general rules of arithmetic and their denotation is simply prefixing the corresponding positive numeral by a minus sign, e.g. −123.456.